[FOM] Godel's First Incompleteness Theorem as it possibly relates to Physics

Brian Hart hart.bri at gmail.com
Mon Oct 13 13:39:15 EDT 2008

On Sat, Oct 11, 2008 at 9:43 AM, Alasdair Urquhart
<urquhart at cs.toronto.edu> wrote:
> On Thu, 9 Oct 2008, Brian Hart wrote:
>> Why doesn't Godel's 1st Incompleteness Theorem imply the
>> incompleteness of any theory of physics T, assuming that T is
>> consistent and uses arithmetic?  Shouldn't the constructors of the
>> Theory of Everything be alarmed?  I know this suggestion of
>> application of Godel's theorem was made decades ago but why didn't it
>> make a bigger impact?  Is it because it is wrong or were there some
>> sociological reasons for mainstream ignorance of it?
> The basic problem with this idea is that it is consistent with
> current knowledge (as far as I know) that there could be a Theory of
> Everything that is in some sense complete in its physical implications,
> though remaining incomplete in its mathematical foundations.

Well, that's the thing: how do you connect the mathematical
incompleteness to its physical analog?  If all of physics is
mathematics, then couldn't an axiomatized physics be completely
founded upon ZFC somehow if one were to follow Hilbert's (other)
Program outlined in his 6th problem?  The axioms of physics would be
relatively primitive, basic physical truths only decomposable within
mathematical axiomatics like ZFC.  If physics were completely founded
upon ZFC then it wouldn't be prone to some form of incompleteness
because incompleteness only rears its head when one considers some
kind of extension of ZFC like ZFC + some large cardinal, correct?

> Of course, it remains rather unclear what we mean by "complete in its
> physical implications."  But I would guess that physicists would be
> very happy with a fundamental theory that predicts all of the
> basic properties of the elementary particles, including the
> constants that currently have to be "put in by hand."
> Of course, gravity would have to be included as well, and that
> seems at the moment to be a very intractable problem.
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